SRC/dtgsna.f(3)

NAME
SRC/dtgsna.f

SYNOPSIS

  Functions/Subroutines
subroutine dtgsna (job, howmny, select, n, a, lda, b, ldb, vl, ldvl, vr, ldvr, s, dif, mm, m, work, lwork, iwork, info)
DTGSNA

Function/Subroutine Documentation

  subroutine dtgsna (character job, character howmny, logical, dimension( * ) select, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( ldvl, * ) vl, integer ldvl, double precision, dimension( ldvr, * ) vr, integer ldvr, double precision, dimension( * ) s, double precision, dimension( * ) dif, integer mm, integer m, double precision, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer info)

DTGSNA

Purpose:

    !>
    !> DTGSNA estimates reciprocal condition numbers for specified
    !> eigenvalues and/or eigenvectors of a matrix pair (A, B) in
    !> generalized real Schur canonical form (or of any matrix pair
    !> (Q*A*Z**T, Q*B*Z**T) with orthogonal matrices Q and Z, where
    !> Z**T denotes the transpose of Z.
    !>
    !> (A, B) must be in generalized real Schur form (as returned by DGGES),
    !> i.e. A is block upper triangular with 1-by-1 and 2-by-2 diagonal
    !> blocks. B is upper triangular.
    !>
    !>

Parameters
    JOB

    !>          JOB is CHARACTER*1
    !>          Specifies whether condition numbers are required for
    !>          eigenvalues (S) or eigenvectors (DIF):
    !>          = 'E': for eigenvalues only (S);
    !>          = 'V': for eigenvectors only (DIF);
    !>          = 'B': for both eigenvalues and eigenvectors (S and DIF).
    !>

    HOWMNY

    !>          HOWMNY is CHARACTER*1
    !>          = 'A': compute condition numbers for all eigenpairs;
    !>          = 'S': compute condition numbers for selected eigenpairs
    !>                 specified by the array SELECT.
    !>

    SELECT

    !>          SELECT is LOGICAL array, dimension (N)
    !>          If HOWMNY = 'S', SELECT specifies the eigenpairs for which
    !>          condition numbers are required. To select condition numbers
    !>          for the eigenpair corresponding to a real eigenvalue w(j),
    !>          SELECT(j) must be set to .TRUE.. To select condition numbers
    !>          corresponding to a complex conjugate pair of eigenvalues w(j)
    !>          and w(j+1), either SELECT(j) or SELECT(j+1) or both, must be
    !>          set to .TRUE..
    !>          If HOWMNY = 'A', SELECT is not referenced.
    !>

    N

    !>          N is INTEGER
    !>          The order of the square matrix pair (A, B). N >= 0.
    !>

    A

    !>          A is DOUBLE PRECISION array, dimension (LDA,N)
    !>          The upper quasi-triangular matrix A in the pair (A,B).
    !>

    LDA

    !>          LDA is INTEGER
    !>          The leading dimension of the array A. LDA >= max(1,N).
    !>

    B

    !>          B is DOUBLE PRECISION array, dimension (LDB,N)
    !>          The upper triangular matrix B in the pair (A,B).
    !>

    LDB

    !>          LDB is INTEGER
    !>          The leading dimension of the array B. LDB >= max(1,N).
    !>

    VL

    !>          VL is DOUBLE PRECISION array, dimension (LDVL,M)
    !>          If JOB = 'E' or 'B', VL must contain left eigenvectors of
    !>          (A, B), corresponding to the eigenpairs specified by HOWMNY
    !>          and SELECT. The eigenvectors must be stored in consecutive
    !>          columns of VL, as returned by DTGEVC.
    !>          If JOB = 'V', VL is not referenced.
    !>

    LDVL

    !>          LDVL is INTEGER
    !>          The leading dimension of the array VL. LDVL >= 1.
    !>          If JOB = 'E' or 'B', LDVL >= N.
    !>

    VR

    !>          VR is DOUBLE PRECISION array, dimension (LDVR,M)
    !>          If JOB = 'E' or 'B', VR must contain right eigenvectors of
    !>          (A, B), corresponding to the eigenpairs specified by HOWMNY
    !>          and SELECT. The eigenvectors must be stored in consecutive
    !>          columns ov VR, as returned by DTGEVC.
    !>          If JOB = 'V', VR is not referenced.
    !>

    LDVR

    !>          LDVR is INTEGER
    !>          The leading dimension of the array VR. LDVR >= 1.
    !>          If JOB = 'E' or 'B', LDVR >= N.
    !>

    S

    !>          S is DOUBLE PRECISION array, dimension (MM)
    !>          If JOB = 'E' or 'B', the reciprocal condition numbers of the
    !>          selected eigenvalues, stored in consecutive elements of the
    !>          array. For a complex conjugate pair of eigenvalues two
    !>          consecutive elements of S are set to the same value. Thus
    !>          S(j), DIF(j), and the j-th columns of VL and VR all
    !>          correspond to the same eigenpair (but not in general the
    !>          j-th eigenpair, unless all eigenpairs are selected).
    !>          If JOB = 'V', S is not referenced.
    !>

    DIF

    !>          DIF is DOUBLE PRECISION array, dimension (MM)
    !>          If JOB = 'V' or 'B', the estimated reciprocal condition
    !>          numbers of the selected eigenvectors, stored in consecutive
    !>          elements of the array. For a complex eigenvector two
    !>          consecutive elements of DIF are set to the same value. If
    !>          the eigenvalues cannot be reordered to compute DIF(j), DIF(j)
    !>          is set to 0; this can only occur when the true value would be
    !>          very small anyway.
    !>          If JOB = 'E', DIF is not referenced.
    !>

    MM

    !>          MM is INTEGER
    !>          The number of elements in the arrays S and DIF. MM >= M.
    !>

    M

    !>          M is INTEGER
    !>          The number of elements of the arrays S and DIF used to store
    !>          the specified condition numbers; for each selected real
    !>          eigenvalue one element is used, and for each selected complex
    !>          conjugate pair of eigenvalues, two elements are used.
    !>          If HOWMNY = 'A', M is set to N.
    !>

    WORK

    !>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
    !>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
    !>

    LWORK

    !>          LWORK is INTEGER
    !>          The dimension of the array WORK. LWORK >= max(1,N).
    !>          If JOB = 'V' or 'B' LWORK >= 2*N*(N+2)+16.
    !>
    !>          If LWORK = -1, then a workspace query is assumed; the routine
    !>          only calculates the optimal size of the WORK array, returns
    !>          this value as the first entry of the WORK array, and no error
    !>          message related to LWORK is issued by XERBLA.
    !>

    IWORK

    !>          IWORK is INTEGER array, dimension (N + 6)
    !>          If JOB = 'E', IWORK is not referenced.
    !>

    INFO

    !>          INFO is INTEGER
    !>          =0: Successful exit
    !>          <0: If INFO = -i, the i-th argument had an illegal value
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  The reciprocal of the condition number of a generalized eigenvalue
    !>  w = (a, b) is defined as
    !>
    !>       S(w) = (|u**TAv|**2 + |u**TBv|**2)**(1/2) / (norm(u)*norm(v))
    !>
    !>  where u and v are the left and right eigenvectors of (A, B)
    !>  corresponding to w; |z| denotes the absolute value of the complex
    !>  number, and norm(u) denotes the 2-norm of the vector u.
    !>  The pair (a, b) corresponds to an eigenvalue w = a/b (= u**TAv/u**TBv)
    !>  of the matrix pair (A, B). If both a and b equal zero, then (A B) is
    !>  singular and S(I) = -1 is returned.
    !>
    !>  An approximate error bound on the chordal distance between the i-th
    !>  computed generalized eigenvalue w and the corresponding exact
    !>  eigenvalue lambda is
    !>
    !>       chord(w, lambda) <= EPS * norm(A, B) / S(I)
    !>
    !>  where EPS is the machine precision.
    !>
    !>  The reciprocal of the condition number DIF(i) of right eigenvector u
    !>  and left eigenvector v corresponding to the generalized eigenvalue w
    !>  is defined as follows:
    !>
    !>  a) If the i-th eigenvalue w = (a,b) is real
    !>
    !>     Suppose U and V are orthogonal transformations such that
    !>
    !>              U**T*(A, B)*V  = (S, T) = ( a   *  ) ( b  *  )  1
    !>                                        ( 0  S22 ),( 0 T22 )  n-1
    !>                                          1  n-1     1 n-1
    !>
    !>     Then the reciprocal condition number DIF(i) is
    !>
    !>                Difl((a, b), (S22, T22)) = sigma-min( Zl ),
    !>
    !>     where sigma-min(Zl) denotes the smallest singular value of the
    !>     2(n-1)-by-2(n-1) matrix
    !>
    !>         Zl = [ kron(a, In-1)  -kron(1, S22) ]
    !>              [ kron(b, In-1)  -kron(1, T22) ] .
    !>
    !>     Here In-1 is the identity matrix of size n-1. kron(X, Y) is the
    !>     Kronecker product between the matrices X and Y.
    !>
    !>     Note that if the default method for computing DIF(i) is wanted
    !>     (see DLATDF), then the parameter DIFDRI (see below) should be
    !>     changed from 3 to 4 (routine DLATDF(IJOB = 2 will be used)).
    !>     See DTGSYL for more details.
    !>
    !>  b) If the i-th and (i+1)-th eigenvalues are complex conjugate pair,
    !>
    !>     Suppose U and V are orthogonal transformations such that
    !>
    !>              U**T*(A, B)*V = (S, T) = ( S11  *   ) ( T11  *  )  2
    !>                                       ( 0    S22 ),( 0    T22) n-2
    !>                                         2    n-2     2    n-2
    !>
    !>     and (S11, T11) corresponds to the complex conjugate eigenvalue
    !>     pair (w, conjg(w)). There exist unitary matrices U1 and V1 such
    !>     that
    !>
    !>       U1**T*S11*V1 = ( s11 s12 ) and U1**T*T11*V1 = ( t11 t12 )
    !>                      (  0  s22 )                    (  0  t22 )
    !>
    !>     where the generalized eigenvalues w = s11/t11 and
    !>     conjg(w) = s22/t22.
    !>
    !>     Then the reciprocal condition number DIF(i) is bounded by
    !>
    !>         min( d1, max( 1, |real(s11)/real(s22)| )*d2 )
    !>
    !>     where, d1 = Difl((s11, t11), (s22, t22)) = sigma-min(Z1), where
    !>     Z1 is the complex 2-by-2 matrix
    !>
    !>              Z1 =  [ s11  -s22 ]
    !>                    [ t11  -t22 ],
    !>
    !>     This is done by computing (using real arithmetic) the
    !>     roots of the characteristical polynomial det(Z1**T * Z1 - lambda I),
    !>     where Z1**T denotes the transpose of Z1 and det(X) denotes
    !>     the determinant of X.
    !>
    !>     and d2 is an upper bound on Difl((S11, T11), (S22, T22)), i.e. an
    !>     upper bound on sigma-min(Z2), where Z2 is (2n-2)-by-(2n-2)
    !>
    !>              Z2 = [ kron(S11**T, In-2)  -kron(I2, S22) ]
    !>                   [ kron(T11**T, In-2)  -kron(I2, T22) ]
    !>
    !>     Note that if the default method for computing DIF is wanted (see
    !>     DLATDF), then the parameter DIFDRI (see below) should be changed
    !>     from 3 to 4 (routine DLATDF(IJOB = 2 will be used)). See DTGSYL
    !>     for more details.
    !>
    !>  For each eigenvalue/vector specified by SELECT, DIF stores a
    !>  Frobenius norm-based estimate of Difl.
    !>
    !>  An approximate error bound for the i-th computed eigenvector VL(i) or
    !>  VR(i) is given by
    !>
    !>             EPS * norm(A, B) / DIF(i).
    !>
    !>  See ref. [2-3] for more details and further references.
    !>

Contributors:
    Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

References:

    !>
    !>  [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
    !>      Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
    !>      M.S. Moonen et al (eds), Linear Algebra for Large Scale and
    !>      Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
    !>
    !>  [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
    !>      Eigenvalues of a Regular Matrix Pair (A, B) and Condition
    !>      Estimation: Theory, Algorithms and Software,
    !>      Report UMINF - 94.04, Department of Computing Science, Umea
    !>      University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working
    !>      Note 87. To appear in Numerical Algorithms, 1996.
    !>
    !>  [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
    !>      for Solving the Generalized Sylvester Equation and Estimating the
    !>      Separation between Regular Matrix Pairs, Report UMINF - 93.23,
    !>      Department of Computing Science, Umea University, S-901 87 Umea,
    !>      Sweden, December 1993, Revised April 1994, Also as LAPACK Working
    !>      Note 75.  To appear in ACM Trans. on Math. Software, Vol 22,
    !>      No 1, 1996.
    !>

Definition at line 378 of file dtgsna.f.

Author

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